Boundedness to a logistic chemotaxis system with singular sensitivity

In this paper, we study the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic-type source: $ u_t=\Delta u-\chi\nabla\cdot(\frac{u}{v}\nabla v)+ru-\mu u^k$,$0=\Delta v-v+u$ under the non-flux boundary conditions in a smooth bounded convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $\chi ,r,\mu >0$, $k>1$ and $n\ge 2$. It is shown that the system possesses a globally bounded classical solution if $k>\frac{3n-2}{n}$, and $r>\frac{{\chi }^2}{4}$ for $0<\chi \le 2$, or $r>\chi -1$ for $\chi >2$. In addition, under the same condition for $r,\chi$, the system admits a global generalized solution when $k\in (2-\frac{1}{n},\frac{3n-2}{n}]$, moreover this global generalized solution should be globally bounded provided $\frac{r}{\mu }$ and the initial data $u_0$ suitably small.